PreprintCC BY 4.0arXiv
On Divisor Function Polynomials and Dirichlet Series
Abstract
We find the Dirichlet series and convergence bounds for the counting function of sum systems (collections of sets of non-negative integers whose Minkowski sum is an arithmetic progression). This motivates the definition and study of divisor function polynomials, which are generating functions for the non-trivial divisor function and for the balanced divisor function that counts, up to a sign, non-trivial ordered factorisations into squarefree factors. These divisor function polynomials turn out to be intimately connected between each other and with Bernoulli numbers and ordered Bell polynomials.
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BibTeX
@misc{huxley2026divisor,
title = {{On Divisor Function Polynomials and Dirichlet Series}},
author = {Martin N. Huxley and Matthew C. Lettington and Karl Michael Schmidt},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13864},
eprint = {2610.13864},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13864},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On Divisor Function Polynomials and Dirichlet Series
AU - Martin N. Huxley
AU - Matthew C. Lettington
AU - Karl Michael Schmidt
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13864
UR - https://arxiv.org/abs/2610.13864
AB - We find the Dirichlet series and convergence bounds for the counting function of sum systems (collections of sets of non-negative integers whose Minkowski sum is an arithmetic progression). This motivates the definition and study of divisor function polynomials, which are generating functions for the non-trivial divisor function and for the balanced divisor function that counts, up to a sign, non-trivial ordered factorisations into squarefree factors. These divisor function polynomials turn out to be intimately connected between each other and with Bernoulli numbers and ordered Bell polynomials.
ER -